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Properties

Label 16.8.14578339124...5625.1
Degree $16$
Signature $[8, 4]$
Discriminant $5^{8}\cdot 29^{6}\cdot 89^{4}$
Root discriminant $24.28$
Ramified primes $5, 29, 89$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $C_4.C_2^2:D_4$ (as 16T211)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![256, -512, -2176, 1248, 5392, 456, -3624, 96, 1985, -21, -689, -51, 137, 22, -16, -3, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 3*x^15 - 16*x^14 + 22*x^13 + 137*x^12 - 51*x^11 - 689*x^10 - 21*x^9 + 1985*x^8 + 96*x^7 - 3624*x^6 + 456*x^5 + 5392*x^4 + 1248*x^3 - 2176*x^2 - 512*x + 256)
 
gp: K = bnfinit(x^16 - 3*x^15 - 16*x^14 + 22*x^13 + 137*x^12 - 51*x^11 - 689*x^10 - 21*x^9 + 1985*x^8 + 96*x^7 - 3624*x^6 + 456*x^5 + 5392*x^4 + 1248*x^3 - 2176*x^2 - 512*x + 256, 1)
 

Normalized defining polynomial

\( x^{16} - 3 x^{15} - 16 x^{14} + 22 x^{13} + 137 x^{12} - 51 x^{11} - 689 x^{10} - 21 x^{9} + 1985 x^{8} + 96 x^{7} - 3624 x^{6} + 456 x^{5} + 5392 x^{4} + 1248 x^{3} - 2176 x^{2} - 512 x + 256 \)

magma: DefiningPolynomial(K);
 
sage: K.defining_polynomial()
 
gp: K.pol
 

Invariants

Degree:  $16$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[8, 4]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(14578339124454047265625=5^{8}\cdot 29^{6}\cdot 89^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $24.28$
magma: Abs(Discriminant(Integers(K)))^(1/Degree(K));
 
sage: (K.disc().abs())^(1./K.degree())
 
gp: abs(K.disc)^(1/poldegree(K.pol))
 
Ramified primes:  $5, 29, 89$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is not Galois over $\Q$.
This is not a CM field.

Integral basis (with respect to field generator \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{8} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{4} a^{10} - \frac{1}{4} a^{9} - \frac{1}{2} a^{8} - \frac{1}{2} a^{7} + \frac{1}{4} a^{6} - \frac{1}{4} a^{5} + \frac{1}{4} a^{4} + \frac{1}{4} a^{3} - \frac{1}{4} a^{2} - \frac{1}{2} a$, $\frac{1}{8} a^{11} - \frac{1}{8} a^{10} - \frac{1}{4} a^{9} + \frac{1}{4} a^{8} - \frac{3}{8} a^{7} - \frac{1}{8} a^{6} - \frac{3}{8} a^{5} - \frac{3}{8} a^{4} + \frac{3}{8} a^{3} - \frac{1}{4} a^{2} - \frac{1}{2} a$, $\frac{1}{16} a^{12} - \frac{1}{16} a^{11} - \frac{1}{8} a^{10} + \frac{1}{8} a^{9} - \frac{3}{16} a^{8} + \frac{7}{16} a^{7} - \frac{3}{16} a^{6} + \frac{5}{16} a^{5} - \frac{5}{16} a^{4} + \frac{3}{8} a^{3} + \frac{1}{4} a^{2}$, $\frac{1}{32} a^{13} - \frac{1}{32} a^{12} - \frac{1}{16} a^{11} + \frac{1}{16} a^{10} - \frac{3}{32} a^{9} - \frac{9}{32} a^{8} + \frac{13}{32} a^{7} - \frac{11}{32} a^{6} + \frac{11}{32} a^{5} - \frac{5}{16} a^{4} + \frac{1}{8} a^{3}$, $\frac{1}{1216} a^{14} - \frac{5}{1216} a^{13} - \frac{7}{608} a^{12} - \frac{31}{608} a^{11} - \frac{83}{1216} a^{10} - \frac{301}{1216} a^{9} - \frac{335}{1216} a^{8} - \frac{135}{1216} a^{7} - \frac{385}{1216} a^{6} - \frac{15}{608} a^{5} - \frac{7}{304} a^{4} + \frac{37}{152} a^{3} + \frac{29}{76} a^{2} + \frac{6}{19} a - \frac{9}{19}$, $\frac{1}{241941666063906688} a^{15} + \frac{72389914790273}{241941666063906688} a^{14} + \frac{835551102995351}{60485416515976672} a^{13} + \frac{592435985454715}{120970833031953344} a^{12} + \frac{245261076552411}{21994696914900608} a^{11} + \frac{8621998879946753}{241941666063906688} a^{10} - \frac{45815640088423221}{241941666063906688} a^{9} + \frac{34174717038546055}{241941666063906688} a^{8} - \frac{39248286215863619}{241941666063906688} a^{7} + \frac{5831824632710193}{60485416515976672} a^{6} - \frac{3452613123955779}{30242708257988336} a^{5} + \frac{2741771241735947}{15121354128994168} a^{4} + \frac{655713371409451}{15121354128994168} a^{3} - \frac{300438379753544}{1890169266124271} a^{2} + \frac{45377687979905}{171833569647661} a + \frac{64199130773026}{1890169266124271}$

magma: IntegralBasis(K);
 
sage: K.integral_basis()
 
gp: K.zk
 

Class group and class number

Trivial group, which has order $1$ (assuming GRH)

magma: ClassGroup(K);
 
sage: K.class_group().invariants()
 
gp: K.clgp
 

Unit group

magma: UK, f := UnitGroup(K);
 
sage: UK = K.unit_group()
 
Rank:  $11$
magma: UnitRank(K);
 
sage: UK.rank()
 
gp: K.fu
 
Torsion generator:  \( -1 \) (order $2$)
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
 
sage: UK.torsion_generator()
 
gp: K.tu[2]
 
Fundamental units:  Units are too long to display, but can be downloaded with other data for this field from 'Stored data to gp' link to the right (assuming GRH)
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 126447.104936 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_4.C_2^2:D_4$ (as 16T211):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 128
The 44 conjugacy class representatives for $C_4.C_2^2:D_4$
Character table for $C_4.C_2^2:D_4$ is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), 4.4.725.1, 4.4.64525.2, 4.4.2225.1, 8.4.15243125.1, 8.4.120740793125.3, 8.8.4163475625.2

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Sibling fields

Degree 16 siblings: data not computed
Degree 32 siblings: data not computed

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type ${\href{/LocalNumberField/2.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/31.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$

In the table, R denotes a ramified prime. Cycle lengths which are repeated in a cycle type are indicated by exponents.

magma: p := 7; // to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
 
magma: idealfactors := Factorization(p*Integers(K)); // get the data
 
magma: [<primefactor[2], Valuation(Norm(primefactor[1]), p)> : primefactor in idealfactors];
 
sage: p = 7; # to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
 
sage: [(e, pr.norm().valuation(p)) for pr,e in K.factor(p)]
 
gp: p = 7; \\ to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
 
gp: idealfactors = idealprimedec(K, p); \\ get the data
 
gp: vector(length(idealfactors), j, [idealfactors[j][3], idealfactors[j][4]])
 

Local algebras for ramified primes

$p$LabelPolynomial $e$ $f$ $c$ Galois group Slope content
$5$5.8.4.1$x^{8} + 10 x^{6} + 125 x^{4} + 2500$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
5.8.4.1$x^{8} + 10 x^{6} + 125 x^{4} + 2500$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
$29$29.4.0.1$x^{4} - x + 19$$1$$4$$0$$C_4$$[\ ]^{4}$
29.4.0.1$x^{4} - x + 19$$1$$4$$0$$C_4$$[\ ]^{4}$
29.8.6.2$x^{8} + 145 x^{4} + 7569$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
$89$89.2.0.1$x^{2} - x + 6$$1$$2$$0$$C_2$$[\ ]^{2}$
89.2.0.1$x^{2} - x + 6$$1$$2$$0$$C_2$$[\ ]^{2}$
89.2.0.1$x^{2} - x + 6$$1$$2$$0$$C_2$$[\ ]^{2}$
89.2.0.1$x^{2} - x + 6$$1$$2$$0$$C_2$$[\ ]^{2}$
89.8.4.1$x^{8} + 427734 x^{4} - 704969 x^{2} + 45739093689$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$